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Yes, it was definitely injured somehow. There are a few things you can do. First of all, don't give baths and let it just soak in the water, as it can introduce bacteria. You can dilute some betadine with water in a jar until it looks like tea and soak just the tail. You can also use neosporin on it. Don't let any stray crickets or roaches stay loose in your beardie's tank, because they can chew on your dragon. Keep an eye on the tail, and if it looks like it's starting to spread up the tail, get it to a reptile vet for treatment. Your beardie will always be missing the very tip, but it should heal ok.
Its not tail rot, but it definately looks like it has been nipped and caused tissue damage. The above advice is great, it should heal fine but it will take time as they heal slowly.